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Investigation on optimal spray properties for groundbased agricultural applications using deposition and
retention models
Nicolas De Cocka,b,∗, Mathieu Massinona, Sofiene Ouled Taleb Salaha,c,d,Frederic Lebeaua
aTERRA Teaching and Research Centre, Gembloux Agro-Bio Tech, University of Liege,B-5030 Gembloux, Belgium
bvon Karman Institute for Fluid Dynamics, B-1640 Rhode-Saint-Genese, BelgiumcCESAM - GRASP, Institute of Physics, University of Liege, Building B5a, Sart Tilman,
B-4000 Liege, BelgiumdTERRA - AgricultureIsLife, Gembloux Agro-Bio Tech, Passage des deportes 2, University
of Liege, B-5030 Gembloux, Belgium
Abstract
In crop protection, it is well know that droplet size determine spray efficacy.
The optimisation of both spray deposition and retention leads to a dilemma:
should small droplets be used to increase retention or large droplets be pre-
ferred to avoid drift? An ideal droplet should have a short time of flight to
minimise its distance travelled while impacting the target with a moderate ki-
netic energy. This paper aims to determine an optimum range of droplet sizes
for boom-sprayer applying herbicide using a modelling approach. The main
parameters of spray deposition and retention models are systematically varied
and the effects on drift potential and droplet impaction outcomes are discussed.
The results of the numerical simulations showed that droplets with diameter
ranging between 200µm and 250µm offer high control of deposition by combin-
ing a low drift potential and a moderate kinetic energy at top of the canopy. A
fourfold reduction of the volume drifting further than 2 m from the nozzle was
observed for a spray with a volume median diameter of 225µm when the relative
span factor of the droplet spectrum was reduced from 1.0 to 0.6. In the latter
∗Corresponding authorEmail address: nicolas.decock@ulg.ac.be (Nicolas De Cock)
Preprint submitted to Journal of LATEX Templates August 3, 2017
scenario, an increase from 63 to 67% of the volumetric proportion of droplets
adhering to the wheat leaf was observed. Therefore, strategies for controlling
the droplet size distribution may offer promising solutions for reducing adverse
impact of spray applications on environment.
Keywords: stochastic Lagrangian; agricultural spray; droplet size
distribution; drift; retention; relative span factor; deposition; retention;
1. Introduction1
Spray application is a key process in crop protection to ensure high yields2
whilst minimising the adverse environmental and health impact of plant protec-3
tion products. During this process, the agricultural mixture is usually atomised4
by passage through a nozzle generating a liquid sheet that further breaks up5
in a cloud of droplets. A herbicide application can be divided in four succes-6
sive stages: deposition (initial spray amount minus off-target losses), retention7
(amount remaining on the plant after impaction), uptake (amount of active in-8
gredient taken into the plant foliage) and translocation (amount of absorbed9
material translocated) (Zabkiewicz, 2007). This paper focuses on deposition10
and retention stages.11
It has been shown that the droplet size distribution of the spray significantly12
affects the deposition (Hilz & Vermeer, 2013; Nuyttens, De Schampheleire,13
Baetens & Sonck, 2007b; Stainier, Destain, Schiffers & Lebeau, 2006; Taylor,14
Womac, Miller & Taylor, 2004).Al Heidary, Douzals, Sinfort & Vallet (2014)15
showed that spray drift decreases with the droplet kinetic energy following a16
power law. Indeed, finer droplets are more prone to drift leading to poten-17
tial product losses in the air, water and soil (Reichenberger, Bach, Skitschak18
& Frede, 2007). Modelling of deposition under field conditions has been real-19
ized using several approaches: Gaussian plume model (Baetens, Ho, Nuyttens,20
De Schampheleire, Melese Endalew, Hertog, Nicolai, Ramon & Verboven, 2009;21
Lebeau, Verstraete, Stainier & Destain, 2011; Raupach, Briggs, Ford, Leys,22
2
Nomenclature
Greek Symbols
β Droplet release angle [◦]
∆t Time step [s]
η, ε Random value from a standard
normal distribution [-]
γ Surface tension [Nm−1]
κ von Karman constant [-]
λ,K Weibull distribution parameter [-]
µ Dynamic viscosity [N s m−2]
ν Kinematic viscosity [m s−2]
ρ Volumetric mass [kg m−3]
σx,z Velocity RMS [m s−1]
τL Lagrangian time scale of turbu-
lence [s]
τ∗L Modified Lagrangian timescale [s]
θ Static contact angle [◦]
Roman Symbols
m Mass flux [kg s−1]
CD Drag coefficient [-]
CDF Cumulative density function [-]
d Droplet diameter [m]
d0 Zero plane displacement [m]
dm Maximum spread diameter [m]
E Arithmetic mean of droplet trav-
eled distance [m]
g Gravity acceleration [m s−2]
hc Crop height [m]
hr Release height [m]
k Liquid to gas dynamic viscosity
ratio [-]
L Monin-Obukhov length [m]
m Droplet mass [kg]
Re Reynolds number [-]
RSF Relative span factor [-]
ToF Time of flight [s]
U Air flow velocity [m s−1]
u Droplet velocity [m s−1]
U∗ Friction velocity [m s−1]
u0 Release velocity [m s−1]
Vr Relative droplet velocity [m s−1]
We Weber number [-]
x Horizontal position [m]
z Vertical position [m]
z0 Surface roughness [m]
Subscripts
g Gaseous
l Liquid
Muschal, Cooper & Edge, 2001), Lagrangian models (Butler Ellis & Miller,23
2010; Holterman, Van De Zande, Porskamp & Huijsmans, 1997; Mokeba, Salt,24
Lee & Ford, 1997; Teske, Bird, Esterly, Curbishley, Ray & Perry, 2002; Walk-25
late, 1987), computational fluid dynamics (CFD) (Baetens, Nuyttens, Verboven,26
De Schampheleire, Nicolai & Ramon, 2007; Weiner & Parkin, 1993). Here a La-27
grangian stochastic model will be used. Lagrangian stochastic models compute28
3
the droplet movement through an airflow using discrete time steps. The airflow29
turbulence is taken into account by superposing a time correlated fluctuating30
component onto a mean component. Dispersal statistics can be retrieved by31
tracking a large number of droplets.32
The amount of spray remaining on a plant after impact is determined by the33
sum of each droplet impact outcomes (adhesion, bounce or shatter). Droplet34
behaviour after impact is mainly governed by droplet kinetic energy, liquid sur-35
face tension and the surface wetability (Josserand & Thoroddsen, 2016; Yarin,36
2006). When a droplet hits a solid surface, it spreads radially producing a thin37
liquid layer. If the droplet kinetic energy at impact overcomes capillary forces,38
the droplet shatters in smaller droplets. Otherwise, the spreading driven by the39
initial kinetic energy of the droplet is decelerated by viscous forces and surface40
tension, until radial dispersion stops. Thereafter, the liquid layer can remain41
pinned on the surface or retract. If the droplet surface energy is sufficient, the42
droplet may detach itself from the surface leading to a bounce (Attane, Girard43
& Morin, 2007). Otherwise, the droplet adheres on the surface. Massinon,44
Dumont, De Cock, Salah & Lebeau (2015) proposed an empirical probabilistic45
model using droplet Weber number to model droplet outcomes on plant leaves.46
Deterministic models of impact outcomes based on energy balance of the impact-47
ing droplet are also available (Mao, Kuhn & Tran, 1997; Mundo, Sommerfeld48
& Tropea, 1995; Dorr, Wang, Mayo, McCue, Forster, Hanan & He, 2015).49
One common approach to reduce drift is to shift the droplet spectrum to-50
wards coarser droplets using low-drift nozzle or by adding spray additives. How-51
ever, coarse droplets present a relatively low degree of surface coverage and may52
bounce or shatter on the target (Hilz & Vermeer, 2013; Massinon, De Cock,53
Forster, Nairn, McCue, Zabkiewicz & Lebeau, 2017). An other solution, is to54
narrow the droplet size distribution towards an intermediate range of droplet55
size.56
The goal of the present paper is to determine an optimum range of droplet57
size for boom-sprayer based herbicide applications using a modelling approach.58
A deposition model based on a stochastic Lagrangian approach is presented in59
4
the section 2.1. The mathematical models determining the droplet outcomes at60
canopy level are presented in the section 2.2. Deposition and retention models61
are used to realise a sensitivity analysis on initial droplet parameters (diameter,62
release height, release velocity) and environmental characteristics (wind speed,63
relative humidity) in the agricultural range detailed in section 2.4.1. Finally, the64
aerial transport and the retention of sprays with different volumetric median65
diameter and relative span factor are assessed in section 2.4.2.66
2. Materials and methods67
2.1. Droplet deposition model68
2.1.1. General overview of the droplet transport model69
Figure 1 shows the flow chart of the model. The simulation starts by ini-70
tialising the droplet characteristics, e.g. its initial location, velocity and size.71
The acceleration and the evaporation of the droplet is then computed at each72
time step. In order to solve the aerodynamic balance of the droplet, the air73
flow characteristics are computed as well at each droplet location taking into74
account atmospheric turbulence. The simulation ends when the droplet either75
looses all its mass or reaches the crop level canopy where the droplet is stated76
to be captured. Air entrainment from the spray nozzle is not taken in account77
in the present model because of a low drop/air mass ratio is assumued which is78
typical of low application volume/high speed applications (Lebeau, 2004).79
2.1.2. Droplet motion80
Equations of droplet motion are taken from the saltation model of Kok81
& Renno (2009), which takes into account the particle inertia. The droplet82
transport model uses a Lagrangian description of the droplet motion. The83
displacement of the droplet after a time t is given by the numerical integration84
of the droplet velocity over time:85
∆xi =
n∑i
ux,i ∆ti & ∆zi =
n∑i
uz,i ∆ti (1)
5
∂mdt
Eq 6
∂udt
Eq 9Eq 2
Uz,iUx,i
Fig. 1: Flow chart of the droplet transport model.
The variation of the droplet velocity is retrieved using Newton’s second law of86
motion taking in account the effects of drag and gravity while neglecting the87
buoyancy.88
∂uxdt =
3CD ρgVr(Ux−ux)4 ρl d
∂uzdt =
3CD ρgVr(Uz−uz)4 ρl d
− g(2)
with m the droplet mass [kg], u the droplet velocity [m s−1], t the time [s], CD89
the drag coefficient [-], ρl and ρg the density of the liquid and gaseous phases90
respectively [kg m−3], A the droplet cross section area [m2], d the droplet diam-91
eter [m], U and u are the air and the droplet velocity respectively [m s−1], Vr is92
the relative velocity between the droplet and the airflow defined as Vr =| u− U |93
[m s−1], and g is the gravitational acceleration rounded to 9.81 [m s−2].94
For Re≤ 400 the drag coefficient of a sphere in a gas flow can be expressed95
using the following expression (Saboni, Alexandrova & Gourdon, 2004):96
Cd =
(k(24Re + 4
Re0.36
)+ 15
Re0.82 − 0.02kRe0.5
1+k
)Re2 + 40 3k+2
Re + 15k + 10
(1 + k)(5 + 0.95Re2)(3)
6
with k equal to the ratio of the liquid to the gas viscosity, k = µlµg
, and Re97
the droplet Reynolds number defined as: Re = Vrdνg
. Other Cd expressions for98
a sphere can be found in the literature (Barati, Neyshabouri & Ahmadi, 2014;99
Langmuir & Blodgett, 1949).100
2.1.3. Description of the air flow101
The velocity profile generated by a wind above crop is made up of a random102
part sum onto a mean component. Assuming the vertical mean flow equal to103
zero, the general formulation is reduced to:104
Ux = Ux + U ′x ; Uz = U ′z (4)
The average part of the horizontal velocity U(zi) is described by a logarith-105
mic velocity profile:106
U(zi) =U∗
κlog
(z − d0z0
)(5)
with κ the von Karman constant equals to 0.41 [-], U∗ the friction velocity107
[m s−1], z the distance above the ground [m], d0 the zero plane displacement108
[m] and z0 the surface roughness [m]. The values d0 and z0 can be related to crop109
height using z0 = 0.1hc and d0 = 0.63hc with hc the crop height (Butler Ellis110
& Miller, 2010).111
For homogeneous isotropic turbulence, the velocity fluctuations U ′ of an air112
particle moving with the flow can be statically described by the following set of113
equations (Kok & Renno, 2009) (Wilson & Sawford, 1996):114
U ′x,i+1 = U ′x,i e− ∆tτ∗Lx + ε σx
√2
(1− e
(−√
∆tτ∗Lx
))
U ′z,i+1 = U ′z,i e− ∆tτ∗Lz + η σz
√2
(1− e
(−√
∆tτ∗Lz
)) (6)
with τ∗L the modified Lagrangian time scale [s], ∆t is the time step [s], η and115
ε are random variables from a standard normal distribution [-], σx and σz are116
the horizontal and the vertical velocity fluctuations [m s−1]. For near neutral117
7
atmospheric conditions: σx = 2.3U∗ and σz = 1.3U∗ (Panofsky, Tennekes,118
Lenschow & Wyngaard, 1977).119
The Lagrangian timescale represents the approximate timescale over which120
the velocities experienced by an air particle are statically related. Since the121
droplets move through the air eddies, the Lagrangian timescale perceived by122
the droplets is shorter. A modified formulation of the Lagrangian timescale for123
the horizontal and the vertical directions was proposed by (Sawford & Guest,124
1991):125
τ∗Lx = τLx√1+(2 Vrσx )
2
τ∗Lz = τLz√1+(Vrσz )
2
(7)
with τL defined as (Butler Ellis & Miller, 2010):126
τL = κU∗(z − d0)
σ2z
√1−
(16(z − d0)
L
)(8)
with L the Monin-Obukhov length [m], which characterises atmospheric stabil-127
ity.128
2.1.4. Droplet evaporation129
Droplet evaporation in the model was based on Guella, Alexandrova &130
Saboni (2008). The set of equations used are described in the Appendix A.131
In this model, the air has a constant vapour fraction and temperature. The loss132
of droplet volume is computed after each time step as:133
∆m =m
ρl∆t (9)
134
2.2. Droplet retention model135
Mathematical models have been developed to predict the outcome of impact-
ing droplets based on an energy balance approach(Dorr et al., 2015; Mao et al.,
8
1997; Mundo et al., 1995). In these models, three impact outcomes are consid-
ered: adhesion, bounce or shatter. Shatter occurs when the inertial forces at
impacting overcome the capillary forces. The droplet shatter threshold may be
predicted based on droplet Reynolds number and Weber number Mundo et al.
(1995):
K = We0.5Re0.25I (10)
Unlike for the drag coefficient, the Reynolds number of the droplet at im-
paction ReI is computed using the liquid kinematic viscosity: ReI = uzdνl
.
The Weber number is expressed as: We =u2z ρl dγ with γ the liquid tension
surface [N m−1]. Experimental measurements have shown that the droplets
shatter when We0.5Re0.25I ≥ Kcrit (Mundo et al., 1995). If the droplet does
not shatter, the model assesses the bounce criteria. Mao et al. (1997) proposed
a semi-empirical model based on energy conservation providing a rebound crite-
ria. Bounce occurs if the excess rebound energy E∗ERE is positive otherwise the
droplet is predicted to adhere to the leaf. The excess rebound energy is defined
as:
E∗ERE =1
4
(dmd
)2
(1−cos θ)+2
3
(d
dm
)−0.12
(dmd
)2.3
(1−cos θ)0.63−1 (11)
with dm the maximum spread diameter [m] and θ the static contact angle [◦].136
The value of dm in the Eq. 11 was, in turn, derived as an implicit function
of We, Re and θ:[0.25 (1− cos θ) + 0.2
We0.83
Re0.33I
](dmd
)3
−(We
12+ 1
)(dmd
)+
2
3= 0 (12)
If there is no real solution for dm in the Eq. 12 or if the computed dm is ≤ d,137
the value of dm is set as equal to d.138
2.3. Numerical procedure139
Figure 2 illustrates the initial state of the simulation. The initial droplet140
location is set as x = 0 and z = hr + hc with zr the release droplet height [m].141
The initial droplet velocity in both directions are: ux = ‖u0‖ cos(β) ; uz =142
‖u0‖ sin(β), with β the angle between the initial droplet direction and the143
9
d
β
hc
hr
z
x
Ux u0
Fig. 2: Initial configuration of the deposition model.
Table 1: Simulation constants. The air and water temperature properties were taken both for
15 ◦C. Subscript g and l refer to gaseous and liquid phases respectively.
Parameter Value Units
µg 1.85e-5 Pa s
µl 1.15e-3 Pa s
ρg 1.2 kg m−3
ρl 1000 kg m−3
hc 0.1 m
L -1000 m
vertical direction [◦] and u0 the release velocity [m s−1]. Liquid and air properties144
used for the computations are shown in the Table 1. The time step ∆t was145
computed as: min(
0.1hVr
, τL10
)[s].146
2.4. Parameter sensitivity study147
2.4.1. Monosized droplets148
A sensitivity analysis was performed to highlight the effect of the droplet di-149
ameter d, wind speed at a height of 2 m U(2), droplet release velocity u0, release150
angle β, the release height above crop hr and relative humidity Hr may have151
on the deposition and retention steps. The variation of these parameters are152
shown in the Table 2. For each instance, the trajectories of 15 000 droplets with153
the same initial conditions were computed. Random wind fluctuations experi-154
10
Table 2: Range of variation of the simulation parameters. The standard values are highlighted
in bold.
Variable Tested values Units
d 100;125;150;175;200;250;300;350;400 µm
U(2) 0;2;4;6;8 m s−1
‖u0‖ 5;10;15 m s−1
β 0;15;30;45;60;75;90 ◦
hr 0.25;0.5;0.75;1 m
Hr 40;60;80 %
enced by the droplets during their flights lead to a variety of trajectories that155
were characterised by statistical parameters such as mean, 5th, 50th (median)156
and 95th percentiles. Later in the paper, if the value of one parameter is not157
specified, the standard values indicated in bold in Table 2 were used.158
The impact outcomes were evaluated on a wheat leaf with water which has a159
static contact angle of 132 ◦ and a Kcrit of 69 (Forster, Mercer & Schou, 2010).160
Water has a surface tension γ of 0.072 N m−1.161
2.4.2. Polydisperse sprays162
The aerial transport of polydisperse sprays of droplets are simulated in order163
to predict the effect of the droplet size distribution on the overall deposition and164
retention. Each spray cloud was simulated by 100 000 droplets randomly drawn165
from a Weibull distribution in volumetric cumulative distribution (CDF) defined166
as: CDF = 1 − e−(−xλ )
K
(Rosin & Rammler, 1933; Babinsky & Sojka, 2002).167
The two Weibull distribution parameters were set to achieve a specific relative168
span factor RSF and volumetric mean diameter Dv50. The relative span factor169
is defined as RSF = Dv90−Dv10Dv50
with Dv10, Dv50 and Dv90 corresponding to the170
maximum droplet diameter below which 10 %, 50 % and 90 % of the volume of171
the sample exists, respectively. Six different values of Dv50 (150, 200, 225, 250,172
300, 350µm) and two RSF (0.6, 1) were simulated resulting in twelve different173
simulations. The twelve simulated droplet size distributions are shown in Fig.3.174
11
0 100 200 300 400 500 600
Diameter [µm]
0
10
20
30
40
50
60
70
80
90
100
Cu
mula
tive v
olu
me
pro
port
ion [
%]
Span = 0.6 VMD = 150 µm
Span = 0.6 VMD = 200 µm
Span = 0.6 VMD = 225 µm
Span = 0.6 VMD = 250 µm
Span = 0.6 VMD = 300 µm
Span = 0.6 VMD = 350 µm
Span = 1.0 VMD = 150 µm
Span = 1.0 VMD = 200 µm
Span = 1.0 VMD = 225 µm
Span = 1.0 VMD = 250 µm
Span = 1.0 VMD = 300 µm
Span = 1.0 VMD = 350 µm
Fig. 3: Cumulative droplet size distribution of the virtual sprays for the six Dv50 and the two
RSF .
Sprays characterised with a RSF of 0.6 and 1 are representative of the nar-175
row spray droplet size distributions produced by rotary atomisers (Qi, Miller176
& Fu, 2008) and flat fan nozzles respectively (De Cock, Massinon, Nuyttens,177
Dekeyser & Lebeau, 2016; Nuyttens, Baetens, De Schampheleire & Sonck, 2007a).178
A Dv50 of 250µm with a RSF of 1 is similar to a spray generated by a flat fan179
nozzle 110-03 operating at at 300 kPa. For all these cases, simulation parameters180
were set to standard values (Table 2).181
3. Results182
3.1. Sensitivity analysis of a population of monodisperse droplets183
3.1.1. Effect of droplet size on velocity dynamics184
Figure 4 a shows the evolution of the vertical median droplet velocity with185
respect to the droplet vertical position. The droplets are released 0.6 m above a186
crop of 0.1 m high with an initial horizontal velocity ux of 0 m s−1 and an initial187
vertical velocity uz of -10 m s−1. The droplets were decelerating in the vertical188
direction approaching their settling velocity whilst, in the horizontal direction,189
the droplets were accelerating towards the wind velocity. The droplets with a190
diameter≥ 250µm reached the crop canopy with a vertical velocity above their191
settling velocity. The median time of flight (ToF) for each droplet size is shown192
12
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Vertical position [m]
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
uz [
m s
-1]
1.7
9 s
0.70 s 0.31 s 0.16 s 0.10 s
0.08 s
100 µm
150 µm
200 µm
250 µm
300 µm
400 µm
a)
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Vertical position [m]
0
0.5
1
1.5
ux [
m s
-1]
100 µm
150 µm
200 µm
250 µm
300 µm
400 µm
Wind profile
b)
Fig. 4: a) Median vertical velocity with respect to the droplet vertical location. The median
time of flight to travel from the release point to the crop top canopy for each droplet size is
indicated above each corresponding line. b) Median horizontal velocity with respect to the
droplet vertical location. The average wind velocity profile defined by the Eq.5 for a reference
wind of 2 m s−1 at 2 m is illustrated by the black curve.
next to each line. The ToF is the time between the droplet release and its193
deposit on the canopy. Droplet ToF is shown decreasing with increasing droplet194
size. The 100µm diameter droplets had, on average, 20 times longer ToF than195
400µm diameter droplets. The ToF ratio between the 250µm and the 400µm196
diameter droplets was around 2.197
Figure 4 b shows the evolution of the horizontal median droplet velocity with198
respect to the droplet vertical position. All droplet sizes reached the top canopy199
level at a horizontal velocity approximately equal to the average wind velocity.200
An overshoot of the wind velocity was observed for larger droplets due to their201
inertia, e.g. 250µm droplets are faster than the wind at z≤ 0.2 m.202
3.1.2. Droplet trajectories203
The random wind fluctuations experienced by the droplets lead to a vari-204
ability of trajectories among the simulations. Figure 5 a shows the 5th, 50th205
(median) and 95th percentile of the trajectories of 15 000 droplets under refer-206
ence conditions (cf Table 2). The median is represented by the solid line. The207
5th and 95th percentile are represented by the left and the right dashed line re-208
spectively. The coarser the droplet, the shorter the horizontal distance travelled209
and the dispersion of the travelled distance. The droplets with diameter larger210
13
10-3
10-2
10-1
100
101
Horizontal position [m]
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Vert
ical positio
n [m
]
100 µm
150 µm
200 µm
250 µm
300 µm
400 µm
a) b)
Fig. 5: a) 5th percentile, median and 95th percentile trajectories for 6 different droplet sizes
under standard conditions. b) Deposition pattern of 15 000 droplets with the same size under
standard conditions. The line with the bullets represents the simulated data and the full lines
represents the log-normal fit. The log-normal distribution arithmetic mean and the arithmetic
variance are displayed above each curves. Details on these parameters are available in the
Appendix B.
than 200µm reached the canopy within 1 m from the release position with a211
dispersion shorter than 0.1 m.212
The 95th percentile curve for the 100µm droplet features a plateau between213
0.1 m and 1 m. This plateau arises from a succession of random velocity fluctu-214
ations directed upwards. At a wind speed of 2 m s−1 at 2 m height, the vertical215
velocity fluctuations are equal to u′z = 0.284 ε with ε a random standard Gaus-216
sian value which is in the same range than the settling velocity of droplet of217
100µm (i.e. 0.29 m s−1). Computations (not displayed here for brevity) showed218
that with a higher wind speeds, the plateau forms a bell shape due to the in-219
crease in the strength of the vertical velocity fluctuations.220
The simulated relative deposition patterns over distance is shown in Fig-221
ure 5 b by dashed with bullets. The full line represents the log-normal fit on222
the simulated data. The fitted and simulated data are in good agreement. The223
next subsection assess the effect of the wind speed and the release parameters on224
the arithmetic mean of the log-normal distribution. The value of the arithmetic225
mean has been retrieved with a least square fitting of the log-normal parameters226
on the numerical data using Matlab (MATLAB 9.0, The MathWorks Inc., Nat-227
14
100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Release height [m]
0
1
2
3
4
5
6 E
[m
] 0.2 0.4 0.6 0.8 1
Release height [m]
0
0.2
0.4
0.6
0.8
1
E [
m]
a)
0 1 2 3 4 5 6 7 8
Wind Speed [m s-1]
0
1
2
3
4
5
6
7
8
9
10
E [
m] 0 2 4 6 8
Wind Speed [m s-1
]
0
0.2
0.4
0.6
0.8
E [
m]
b)
5 6 7 8 9 10 11 12 13 14 15
Release velocity [m s-1]
0
0.5
1
1.5
2
2.5
3
E [
m]
c)
0 10 20 30 40 50 60 70 80 90
Release angle β [°]
0
0.5
1
1.5
2
2.5
3
E [
m]
d)
Fig. 6: Effect of the release height, wind speed, release velocity and release angle on the
average of the log-normal fit arithmetic mean E.
ick, MA, USA). More details about the log-normal distribution and the reduced228
parameters are furnished in the Appendix B.229
3.1.3. Average droplet transport230
The results of the arithmetic mean E [m] with respect to variation of the231
ejection height, ejection angle, wind speed and ejection velocity are presented232
in Fig.6. The horizontal distance travelled by a droplet was correlated with233
droplet ToF and wind speed. ToF decreased with decreasing release velocity234
and increasing droplet settling velocity. The release height increased the aver-235
age displacement, mainly for droplet smaller than 250µm. For fine droplets, the236
release height was roughly proportional to the ToF since the droplets quickly237
15
reached their settling velocities, leading to a linear relationship between trav-238
elled distance and release height. For droplets coarser than 200µm, the latter239
relationship was not linear because larger droplets adecelerate during their fall.240
The travelled distance linearly increased with increasing wind speed. Finer241
droplets were more sensitive to wind speed, resulting in steeper slopes in the242
graph of Fig.6 b. Increase in the release velocity slightly decreased the traveled243
distance for the finer droplets (- 20% for 100µm 5-15 m s−1) whilst the decrease244
was substantial for coarse droplets (- 80% for 400µm 5-15 m s−1) which relates245
to droplet inertia.The effect of the release angle β is shown in Fig.6 d. For each246
angle, the average displacement without wind was subtracted to consider the247
effect of these angles. The increase of β leads to a decrease in initial vertical248
velocity and an increase of the initial horizontal velocity, increasing the averaged249
travelled distance. Droplets with diameter ≥ 200µm had an average horizontal250
displacement shorter than 0.5 m for release angle ≤ 60◦.251
3.1.4. Droplet transport of 95th percentile252
X95 represents the downwind distance by which 95 % of the spray volume has253
reached the ground. It corresponds to the final position of the 95th percentile254
trajectories shown in Fig.5 a. This parameter was responsive to the average255
transport of the droplet spray and deposition dispersion. The effect of the main256
parameters on the X95 is shown in Fig.7. The increase of release height was lin-257
ear with increasing X95, similarly to Fig.6 a. However, the decrease of the slope258
with increasing droplet size was stronger than for the average displacement since259
the increase of height also enhanced the deposition variability. The increase of260
wind speed generated a quadratic increase of the X95. This can be explained by261
an increase in the random wind fluctuations which in turn enhanced the vari-262
ability of the droplet trajectories. Therefore, the log-normal curves representing263
the volume distribution over distance were strongly flattened.At windspeeds of264
8 m s−1, more than 5 % of the droplets of 100µm travelled further than 100 m.265
This distance dropped below 1 m for droplets with diameter ≥ 250µm. The266
increase of the release velocity led to a moderate decrease of X95. Thus, acting267
16
100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Release height [m]
0
2
4
6
8
10
12
14
16
18
20
X9
5 [
m]
0.2 0.4 0.6 0.8 1
Release height [m]
0
0.5
1
1.5
2
2.5
3
X9
5 [m
]
a)
0 1 2 3 4 5 6 7 8
Wind Speed [m s-1]
0
20
40
60
80
100
120
X95 [m
]
0 2 4 6 8
Wind Speed [m s-1
]
0
1
2
3
4
5
6
7
X95 [m
]
b)
5 6 7 8 9 10 11 12 13 14 15
Release velocity [m s-1]
0
1
2
3
4
5
6
7
X95 [
m]
c)
0 10 20 30 40 50 60 70 80 90
Release angle β [°]
0
1
2
3
4
5
6
7
X95 [
m]
d)
Fig. 7: Effect of the release height, wind speed, release velocity and release angle on the
distance above which 95 % of the droplets have reach the top canopy level X95.
17
100 150 200 250 300 350 400
Droplet diameter [µm]
0
5
10
15
uz a
t cro
p c
an
op
y [
m s
-1]
Shatter limit
Bounce limit
0.25 m at 5 m s -1
0.25 m at 10 m s -1
0.25 m at 15 m s -1
0.50 m at 5 m s -1
0.50 m at 10 m s -1
0.50 m at 15 m s -1
0.75 m at 5 m s -1
0.75 m at 10 m s -1
0.75 m at 15 m s -1
Settling velocity
a)
100 150 200 250 300 350 400
Droplet diameter [µm]
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
An
gle
with
th
e h
orizo
nta
l d
ire
ctio
n [
°]
0.25 m at 5 m s -1
0.25 m at 10 m s -1
0.25 m at 15 m s -1
0.50 m at 5 m s -1
0.50 m at 10 m s -1
0.50 m at 15 m s -1
0.75 m at 5 m s -1
0.75 m at 10 m s -1
0.75 m at 15 m s -1
b)
Fig. 8: a) Average impact velocity in respect to the droplet diameter for three release heights
and three release velocities. The dashed line shows the velocity above which a droplet would
shatter or bounce while impact a wheat leaf using the adhesion model described in section 2.2.
b) Average droplet trajectory angle at the crop top canopy level with the horizontal direction.
The other simulation parameters were set at standard values (cf Table 2).
solely on the release velocity does not significantly affect drift. The increase of268
β led to an increase of X95, especially for coarser droplets at release angles from269
60◦ to 90◦. At an angle of 60◦, less than 5 % of the droplets with diameter ≥270
200µm were airborne further than 1 m. X95 was strongly influence by droplet271
size due to the higher deposition variability and higher average displacement272
for finer droplets. This means that droplets diameter ≤ 150µm should be min-273
imised within the spray since a significant proportion will travel several metres.274
E and X95 were close to each other for droplets with diameter ≥ 250µm showing275
a low dispersion of droplet trajectories. This low dispersion can be explained276
by their shorter ToF relatively to finer droplets.277
3.1.5. Droplet velocity at top canopy level278
Figure 8 a shows the average droplet vertical velocity at crop height for three279
release heights and three release velocities. The coarser the droplet, the shorter280
is the travelled distance and the faster it may impact on the target. The black281
line shows experimental measurements of settling velocities (Gunn & Kinzer,282
1949). At a release height of 0.5 m, droplets smaller with diameter ≤ 200µm283
reached their settling velocity. At 350µm diameter, the 0.25 m and 10 m s−1284
18
line crosses the 0.5 m 15 m s−1 line showing that the increase of release velocity285
overcomes the increase in flight distance for droplets with higher inertia. The286
black dashed line shows the thresholds for droplet bounce and shatter on a287
wheat leaf predicted for water (Dorr et al., 2015). For the whole range of288
droplet size studied, shatter occurs when the droplets move faster than their289
settling velocity. For standard simulation conditions (i.e. 0.5 m and 10 m s−1)290
droplets larger than 400µm shattered and droplets between 270 and 400µm291
bounced. The mitigation of bounce and shatter can be done by increasing the292
release height or by decreasing the release velocity. Nevertheless, decreasing the293
release velocity was predicted as being less detrimental for the spray drift as294
shown in Fig.6 a,c.295
Droplet trajectory at the top canopy affects the potential droplet reten-296
tion. For graminicide application, vertical trajectories reduce the droplet cap-297
ture probability by the target (Jensen, 2012; Spillman, 1984). Figure 8 b shows298
the average trajectory angles in respect with the horizontal direction under a299
wind of 2 m s−1 at 2 m above the crop. The droplet ToF and the droplet size300
will affect the final horizontal velocity whilst the droplet size, release height301
and the release velocity will determine the final vertical velocity. The fine and302
therefore slow droplets reach the canopy more horizontally than the coarse ones.303
For a release height of 0.5 m, the droplets reached the top of the canopy with304
roughly the same horizontal velocity as shown in Fig.4 b. Therefore, the differ-305
ence in angle between droplet size may be mainly related to the vertical velocity306
component.307
3.1.6. Droplet evaporation308
The effect of the relative humidity Hr, wind speed and droplet size on the309
evaporated fraction is shown in Fig.9. The evaporated fraction was computed310
by subtracting the volume of liquid reaching the top canopy from the initial311
volume released. Evaporation mainly affects droplets with diameter ≤ 150µm.312
Droplets with diameter ≥ 250µm had moderate evaporation, i.e. ≤ 3% for the313
worst scenario. Therefore, for droplet ≥ 250µm diameter, the evaporation may314
19
100 µm 125 µm 150 µm 175 µm 200 µm 250 µm 300 µm 350 µm 400 µm
40 45 50 55 60 65 70 75 80
HR [%]
0
10
20
30
40
50
60
70
80
90
100
Vo
lum
e e
vap
ora
te [
%]
a)
0 1 2 3 4 5 6 7 8
Wind Speed [m s -1 ]
0
10
20
30
40
50
60
70
80
90
100
Vo
lum
e e
va
po
rate
[%
]
b)
Fig. 9: Evolution of the relative volume evaporate in respect to the relative humidity and the
wind speed.
not be a concern. The evaporation model does not take into account the small315
increase in vapour pressure in the surrounding air due to the droplet evaporation.316
Therefore the evaporation rate observed in real conditions could be lower.317
3.2. Polydisperse sprays318
3.2.1. Deposition319
Figure 10 a shows the volume of spray airborne with respect to the distance320
from the nozzle for twelve simulations with different Dv50 and RSF . As ex-321
pected, increasing Dv50 reduces the volume of airborne spray. Increasing Dv50322
from 150µm to 350µm reduces the airborne spray at 2 m from 20 % to 2 %.323
Comparison of sprays with the same Dv50 shows that lower the RSF can re-324
duce drift. For a Dv50 of 150µm, decreasing of RSF from 1 to 0.6 produces a325
reduction from 20 % to 12 % of the airborne spray at 2 m. Table 3 summarises326
the airborne spray reduction at several distances induced by reducing the RSF327
from 1.0 to 0.6 computed as: 100Drift 0.6
Drift 1.0. The drift reduction produced by328
the RSF reduction increased with the Dv50 because the coarser the spray, the329
greater the relative reduction of the fine droplets. For the spray with a Dv50330
of 250µm, drift reduction was around 80 % which corresponds to a three star331
rating in the LERAP scheme (Butler Ellis, Alanis, Lane, Tuck, Nuyttens &332
20
Table 3: Airborne spray reduction [%] induced by a RSF reduction from 1.0 to 0.6. The
airborne spray reduction is given for each Dv50 at 5 distances from the release point.
Distance [m]
2 4 6 8 10
Dv50
[µm
]
150 43.4 56.0 55.7 49.2 42.9
200 67.3 76.1 76.0 73.5 70.8
225 74.2 81.3 81.2 79.9 77.2
250 80.3 85.4 85.4 83.5 80.4
300 87.0 90.6 90.0 88.3 87.0
350 90.6 93.1 93.4 93.1 92.7
van de Zande, 2017). For each Dv50 the drift reduction appeared to be roughly333
constant over distance.334
3.2.2. Retention335
Figure 11 shows the relative volume of each droplet impact for the twelve336
simulated sprays on a wheat leaf. For a given RSF , the increase of Dv50 leads337
to a monotonic decrease of adhesion and the emergence of bounce and shatter338
due to a progressive increase of larger droplet proportion. Reduction of RSF339
enhanced one outcome according to the Dv50, for Dv50 ≤ 250µm there was an340
increase of the adhesion whilst bounce increased for Dv50 ≥ 300µm. For stan-341
dard conditions, the diameter threshold between adhesion and bounce is around342
270µm as shown in Fig.8 a. Therefore, a RSF reduction may be detrimental343
if the Dv50 is not in the adequate range as has already been noted in previous344
theoretical work (Massinon, De Cock, Ouled Taleb Salah & Lebeau, 2016).345
4. Discussion346
The study of the droplet transport dynamic has shown that the size of a347
droplet affects its trajectory. The finer the droplet longer its time in the air348
making it more sensitive to evaporation and drift. The droplet ToF can be349
21
0 2 4 6 8 10 12 14 16 18 20
Distance [m]
10-4
10-3
10-2
10-1
100
101
102
Spra
y a
irbone [%
]
Span = 0.6 VMD = 150 µm
Span = 0.6 VMD = 200 µm
Span = 0.6 VMD = 225 µm
Span = 0.6 VMD = 250 µm
Span = 0.6 VMD = 300 µm
Span = 0.6 VMD = 350 µm
Span = 1.0 VMD = 150 µm
Span = 1.0 VMD = 200 µm
Span = 1.0 VMD = 225 µm
Span = 1.0 VMD = 250 µm
Span = 1.0 VMD = 300 µm
Span = 1.0 VMD = 350 µm
Fig. 10: Volume of airborne spray in respect with the distance. Twelve sprays were simulated
with different Dv50 and RSF values. The outcomes have been determined at the top canopy
level using the models described in the section 2.2 for a release height of 0.5 m and a release
speed of 10 m s−1.
22
0
0.2
0.4
0.6
0.8
1
Rela
tive v
olu
me
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Rela
tive v
olu
me
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
150 200 225 250 300 350
0.6
1.0
Dv50 [µm]
RS
F [-]
A SB A SB A SB A SB A SBA SB
A SB A SB A SB A SB A SBA SB
Fig. 11: Droplet impact outcome predictions at top canopy level expressed in relative volume.
A, B and S correspond to adhesion, bounce and shatter respectively.
shortened by decreasing the release height or by increasing the release veloc-350
ity. Release angle and release velocity have moderate effects on fine droplets.351
However for coarse droplets, increasing the release velocity increases the droplet352
velocity at the canopy and thus its outcome during impaction. The change of353
the release angle from vertical to horizontal direction leads to an increase in354
the travelled distance arising from both the initial horizontal velocity and the355
increase in ToF. An optimum value may be around 60◦. The wind speed is356
enhances the average droplet travelled distance linearly and the maximum dis-357
tance quadratically. However, with droplets of 250µm, and 8 m s−1 wind speed358
95 % of the spray reached canopy top level below 1 m from the release point. The359
shatter threshold on a wheat leaf was reached by droplets larger than 400µm360
when the release height is at 0.5 m and the release speed at 10 m s−1. With361
tthese initial conditions, bounce occurs for droplet between 270 and 400µm.362
Therefore, droplets with a diameter from 200 to 270µm have a low drift poten-363
tial and may not shatter or bounce on a wheat leaf.364
23
For a polydisperse sprays, the overall behaviour can be seen as the combi-365
nation of drop size distribution and the properties of each droplet size. Drift366
and the volume of droplet adhesion decrease with increasing Dv50. Narrowing367
the RSF of the spray may solve this problem. A spray with a Dv50 of 225µm368
and a RSF of 0.6 released at 0.5 m at 10 m s−1 above the crop produces low369
drift with moderate kinetic energy at the crop canopy level. Using a Weibull370
distribution, this spray would have a Dv10 of 152µm, Dv90 of 288µm with 1.4 %371
of the droplet volume ≤ 100µm diameter and 9.5 % ≤ 150µm diameter. The372
narrowing of the spray drift may be detrimental when the Dv50 is too small or373
too large which would enhanced drift or decreases retention.374
5. Conclusion375
A combined Lagrangian droplet transport and retention models has been376
presented. The deposition over distance had a log-normal distribution with a377
dispersion and average distance larger for finer droplets. The results of numerical378
simulations showed that droplets with diameters ranging between 200µm and379
250µm offered high control of deposition by combining a low drift potential and380
moderate kinetic energy at the top of the canopy. The reduction of the RSF381
from 1.0 to 0.6 is an effective way to mitigate deposition and retention losses.382
A fourfold reduction of the drift volume at a distance of 2 m from the nozzle383
was observed for a spray with a Dv50 = 225µm when the RSF was reduced384
from 1.0 to 0.6. Under this scenario, an increase in the volumetric proportion385
of adhering droplets on a wheat leaf from 63 to 78% was shown. Therefore,386
strategies to control the droplet size distribution in terms of Dv50 and RSF387
may offer promising solutions for reducing adverse impacts on environment of388
spray applications.389
Further work should be carried out on the experimental assessment of the390
performance of such sprays in term of drift reduction and retention on target.391
Sprays with a RSF around 0.6 and a Dv50 of 225µm appear to feasible using392
rotary atomisers (Qi et al., 2008).393
24
Acknowledgments394
This work was supported by the Fonds De La Recherche Scientifique - FNRS,395
under the FRIA grant n◦ 97 364.396
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29
Appendix A527
The mass flux is given by:528
m = AShgDg
dρg(Yv,s − Ys,∞) (A.1)
with A the droplet area [m2], d the droplet diameter, Shg the gaseous Sher-529
wood number [-], Dg the molecular diffusion [m s−2], Yv,s and Yv,∞ are the vapor530
mass fractions at the droplet interface and far from the droplet respectively [-].531
For a diameters less than 5 mm, Shg can be computed as:532
Shg = 1.61 + 0.718Re0.5Sc0.33g (A.2)
with Scg the Schmidt number for the gaseous phase [-] expressed by: Scg =533
νDg
.534
The molar fraction at the droplet surface Yv,s is computed as:535
Yv,s = yvlMl
Mt(A.3)
with yvl = PsatPtot
and Mt = yvlMl + (1 − yvl )Mg. Ml and Mg are the molar536
mass of the liquid and the gaseous phase respectively [g mol−1]. Therefore, at537
atmospheric pressure the vapour mass fraction at the droplet interface neigh-538
bourhood, Yv,s, reads:539
Yv,s =Psat
Psat + (Ptot − Psat) Mg
Ml
(A.4)
The vapour pressure in the far field is computed using the relative humidity540
Hr:541
Yinf = HrPsat
HrPsat + (Ptot −HrPsat)(Mg
Ml
) (A.5)
The droplet exchanges heat with the air by convection. The heat flux be-542
tween the droplet surface and the surrounding air Qd [J s−1] reads:543
Qd = SlNugλgd
(Tinf − Tl) (A.6)
30
with Tl, Tinf the temperature of the droplet and far from the droplet inter-544
face respectively [K], λg the thermal conductivity of the gaseous phase [W m−1 K−1].545
The gaseous Nusselt number Nug [-] is a function of the gaseous Reynolds and546
the gaseous Prandlt number Prg [-]:547
Nug = 1.61 + 0.718√Re 3√Prg (A.7)
with Prg =Cpµgλg
548
The thermal balance is given by difference between the convection heat flux549
and the latent heat flux:550
Ql = Qd − mLv (A.8)
with Lv the vaporisation latent heat of water [J kg−1] can be expressed as a551
function of the reduced temperature Tr:552
Lv = 52.05310e6 (1− Tr)0.3199−0.212Tr+0.25795T 2r (A.9)
with Tr = Tl+273Tc
for water Tc=647.13 K.553
The temperature at each time step is retrieved by integrate:
VlρlCp∂dTldt
= SlNugλgd
(Tinf − Tl)− Lvm (A.10)
Cp the heat capacity [J K−1]. For water, the heat capacity is given by:554
Cp = 276730− 2090.1T + 8.125T 2 − 0.014116T 3 + 9.3701e6T 4 (A.11)
Appendix B555
The cumulative distribution function (CDF) of a log-normal distribution is
defined as:
CDF =1
xσn√
2πe− (ln(x)−µn)2
2σ2n (B.1)
31
with σn and µn the two log-normal distribution parameters. From these param-556
eters reduced variables can be extracted: the arithmetic mean E = eµn+0.5σ2n557
and the arithmetic variance Var = e2µn+2σ2n
(eσ
2n − 1
).558
32
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